Short answers, pulled from the story.
The ecclesiastical court sentenced men to death for adultery, but secular authorities carried out the execution. This created a legal duality where the church held the power to condemn but not to kill.
The Court of Probate Act 1857 transferred jurisdiction over the personal estates of the deceased to the Court of Probate in the 19th century. This act removed the power of the Ecclesiastical Courts of England to handle these estates while they retained control over church property and disciplinary proceedings.
Appeals from the Diocese of Springfield, Massachusetts go to the tribunal of the Archdiocese of Boston. Cases from the Archdiocese of Boston then go to the Archdiocese of New York, ensuring no case is decided by the same diocese twice.
The Apostolic Penitentiary deals exclusively with matters of the internal forum involving confidential and secret matters like the Sacrament of Penance. It has no jurisdiction in the external forum and focuses on private facts that arise only within the confessional.
The Methodist Judicial Council consists of nine members, both laity and clergy, elected by the General Conference for an eight-year term. The ratio of laity to clergy alternates every four years.